<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01//EN"
   "http://www.w3.org/TR/html4/strict.dtd">
<html>
  <head>
    <title>
      Addenda and errata for papers on geodesics
    </title>
    <meta name="description" content="Geodesics on an ellipsoid,
				      Addenda and Errata" />
    <meta name="author" content="Charles F. F. Karney" />
    <script type="text/javascript"
	    src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
    </script>
  </head>
  <body topmargin=10 leftmargin=10>
    <a href="geod.html">Back to resource page for geodesics.</a>
    <h3>
      <a name="geodalg-addenda">Addenda</a> for C. F. F. Karney,
      <a href="geod.html">
	<i>Algorithms for Geodesics</i></a>,
      <a href="https://dx.doi.org/10.1007/s00190-012-0578-z">
	J. Geodesy <b>87</b>(1), 43&ndash;55 (Jan. 2013)</a>;<br>
      DOI: <a href="https://dx.doi.org/10.1007/s00190-012-0578-z">
	10.1007/s00190-012-0578-z</a>.
    </h3>
    <ol>
      <li>
	Implementations of the geodesic routines are now available in C,
	Fortran, Java, JavaScript, Python, and Matlab (in addition to
	C++).  For details see
	<a href="http://geographiclib.sourceforge.net/html/other.html">
	  this link</a>.
      <li>
	Care needs to be taken when solving the inverse problem for a
	non-equatorial geodesic when both end points on the equator.  In
	Table 5, the quadrants for \(\sigma_1\) and \(\omega_1\) must be
	determined taking \(\sin\sigma_1\) and \(\sin\omega_1\) to be
	negative; this is consistent with the ordering \(\phi_1 \le 0\).
      <li>
	The 6th-order series given in the paper provide solutions for
	the geodesic problem which are accurate to round off for
	\(\left|f\right| \lt 0.01\).  The least accurate of the series
	is the reverted series for \(\sigma\) in terms of \(\tau\),
	Eqs. (20) and (21), which is used only in solving the direct
	problem.  The accuracy can be improved by using these equations
	to give an initial approximation for \(\sigma\) which is
	following by one step of Newton's method applied to Eq. (7),
	with \(dI_1(\sigma)/d\sigma = \sqrt{1 + k^2 \sin^2\sigma}\).
	With this change (which need only be applied for
	\(\left|f\right| \gt 0.01\)), the 6th-order series are accurate
	to round-off for \(\left|f\right| \lt 0.02\).
      <li>
	Equation (63) give expansions for the area integral using
	\(e'^2\) and \(k^2\) as small parameters.  Unfortunately the
	resulting series diverges for \(e' \gt 1\) or \(b/a \lt
	1/\sqrt2\).  This problem is remedied by expanding instead in
	terms of \(n\) and \(\epsilon\).  Thus Eq. (63) becomes
	\[
\begin{align}
C_{40} &\textstyle= \bigl(\frac{2}{3} - \frac{4}{15} n + \frac{8}{105} n^2 +
      \frac{4}{315} n^3 + \frac{16}{3465} n^4 + \frac{20}{9009} n^5\bigr)\\
      &\textstyle\quad{}- \bigl(\frac{1}{5} - \frac{16}{35} n +
      \frac{32}{105} n^2 - \frac{16}{385} n^3 -
      \frac{64}{15015} n^4\bigr) \epsilon\\
      &\textstyle\quad{}- \bigl(\frac{2}{105} + \frac{32}{315} n -
      \frac{1088}{3465} n^2 + \frac{1184}{5005} n^3\bigr) \epsilon^2\\
      &\textstyle\quad{}+ \bigl(\frac{11}{315} - \frac{368}{3465} n -
      \frac{32}{6435} n^2\bigr) \epsilon^3\\
      &\textstyle\quad{}+ \bigl(\frac{4}{1155} +
      \frac{1088}{45045} n\bigr) \epsilon^4
      + \frac{97}{15015} \epsilon^5 + \ldots,\\
C_{41} &\textstyle=  \bigl(\frac{1}{45} - \frac{16}{315} n +
      \frac{32}{945} n^2 - \frac{16}{3465} n^3 -
      \frac{64}{135135} n^4\bigr) \epsilon\\
      &\textstyle\quad{}- \bigl(\frac{2}{105} - \frac{64}{945} n +
      \frac{128}{1485} n^2 - \frac{1984}{45045} n^3\bigr) \epsilon^2\\
      &\textstyle\quad{}- \bigl(\frac{1}{105} - \frac{16}{2079} n -
      \frac{5792}{135135} n^2\bigr) \epsilon^3\\
      &\textstyle\quad{}+ \bigl(\frac{4}{1155} -
      \frac{2944}{135135} n\bigr) \epsilon^4
      + \frac{1}{9009} \epsilon^5 + \ldots,\\
C_{42} &\textstyle=  \bigl(\frac{4}{525} - \frac{32}{1575} n +
      \frac{64}{3465} n^2 - \frac{32}{5005} n^3\bigr) \epsilon^2\\
      &\textstyle\quad{}- \bigl(\frac{8}{1575} - \frac{128}{5775} n +
      \frac{256}{6825} n^2\bigr) \epsilon^3\\
      &\textstyle\quad{}- \bigl(\frac{8}{1925} -
      \frac{1856}{225225} n\bigr) \epsilon^4
      +  \frac{8}{10725} \epsilon^5 + \ldots,\\
C_{43} &\textstyle=  \bigl(\frac{8}{2205} - \frac{256}{24255} n +
      \frac{512}{45045} n^2\bigr) \epsilon^3\\
      &\textstyle\quad{}- \bigl(\frac{16}{8085} -
      \frac{1024}{105105} n\bigr) \epsilon^4
      - \frac{136}{63063} \epsilon^5 + \ldots,\\
C_{44} &\textstyle=  \bigl(\frac{64}{31185} -
      \frac{512}{81081} n\bigr) \epsilon^4
      - \frac{128}{135135} \epsilon^5 + \ldots,\\
C_{45} &\textstyle=  \frac{128}{99099} \epsilon^5 + \ldots.
\end{align}
	\]
      <li>
	In some applications, it is necessary to keep track of how many
	times a geodesic encircles the earth, i.e., to determine the
	value of \(\lambda_{12}\) without reducing it to some canonical
	range.  The geodesic classes offer this option through the
	Geodesic::LONG_UNROLL mask bit.  In the case of the inverse
	problem, we are interested in the shortest geodesic and thus
	\(\lambda_{12}\in(-\pi,\pi]\) (the geodesic between points on
	opposite meridians is taken to be east-going).  When solving the
	direct geodesic problem, \(\sigma_{12}\) is found in terms of
	the length of the geodesic, \(\omega\) and \(\sigma\) pass from
	one quadrant to the next at the same time, and they are related
	by \(\tan\omega = \sin\alpha_0 \tan\sigma\).  The "unrolled"
	value of \(\lambda_{12}\) is then given by
	\[
	\begin{align}
	\omega_{12} &= E\biggl[\sigma_{12}
	- \biggl(\tan^{-1}\frac{\sin\sigma_2}{\cos\sigma_2} -
	\tan^{-1}\frac{\sin\sigma_1}{\cos\sigma_1}\biggr)\\
	&\quad\qquad{}+ \biggl(\tan^{-1}\frac{E\sin\omega_2}{\cos\omega_2} -
	\tan^{-1}\frac{E\sin\omega_1}{\cos\omega_1}\biggr)\biggr],\\
	\lambda_{12} &= \omega_{12} - f\sin\alpha_0
	\bigl(I_3(\sigma_2)-I_3(\sigma_1)\bigr),
	\end{align}
	\]
	where \(E=\pm1\) is the sign of \(\sin\alpha_0\) or \(+1\) if
	\(\sin\alpha_0 = 0\).
      <li>
	The starting guesses for Newton's method given in Sec. 5 are not
	very good for highly eccentric ellipsoids.  It's possible to
	modify Newton's method so that it converges even for poor
	initial guesses.  The goal is to find the root of
	\[
	f(\alpha_1) \equiv \lambda_{12}(\alpha_1) - \lambda_{12} = 0,
	\]
	where \(\lambda_{12}(\alpha_1)\) is the solution of the hybrid
	problem.  There is exactly one root to this equation in the
	interval \(\alpha_1 \in (0,\pi)\) and its derivative
	\(f'(\alpha_1)\) is positive at the root.  During the course of
	the iteration, a range \((\alpha_{1a}, \alpha_{1b})\) is
	maintained which brackets the root and with each evaluation of
	\(f(\alpha_1)\) the range is shrunk, if possible.  Newton's
	method is restarted whenever the derivative of \(f(\alpha_1)\)
	is negative (because the new value of \(\alpha_1\) is then
	further from the solution) or if the new estimate of
	\(\alpha_1\) lies outside \((0,\pi)\); in this case, the new
	starting guess is taken to be \((\alpha_{1a} + \alpha_{1b}) /
	2\).
      <li>
	In order to obtain accurate solutions for ellipsoids of
	arbitrary eccentricity, it is necessary to replace the series
	expansions for the integrals (which are valid only if \(f\) is
	small) with direct evaluation in terms of elliptic integrals
	(which are valid for all \(f\)).  The key relations used are
	\[
\begin{align}
  \frac sb &= E(\sigma, ik), \\
  \lambda &= \chi
           - \frac{e'^2}{\sqrt{1+e'^2}}\sin\alpha_0 H(\sigma, -e'^2, ik), \\
  J(\sigma) &= k^2 D(\sigma, ik),
\end{align}
	\]
	where
	\[
\begin{align}
\tan\chi &= \sqrt{\frac{1+e'^2}{1+k^2\sin^2\sigma}}\tan\omega, \\
 H(\phi, \alpha^2, k)
 &=
 \frac1{\alpha^2} F(\phi, k) +
      \biggl(1 - \frac1{\alpha^2}\biggr) \Pi(\phi, \alpha^2, k),
\end{align}
	\]
	and \(F(\phi, k)\), \(E(\phi, k)\), \(D(\phi, k)\), and
	\(\Pi(\phi, \alpha^2, k)\), are incomplete elliptic integrals
	(see <a href="http://dlmf.nist.gov/19.2.ii">
	http://dlmf.nist.gov/19.2.ii</a>).
      <li>
	Google Books does not consistently provide access to the full
	text.  If you encounter this situation, you can download the
	pdf files listed here:
	<ul>
	  <li>
	    Gauss (1828),
	    Google id:
	    <a href="https://books.google.com/books?id=a1wTJR3kHwUC">
	      a1wTJR3kHwUC</a>,
	    pdf:
	    <a href="http://geographiclib.sf.net/geodesic-papers/gauss28-en.pdf">
	      gauss28-en.pdf</a>.
	  <li>
	    Helmert (1880),
	    Google id:
	    <a href="https://books.google.com/books?id=qt2CAAAAIAAJ">
	      qt2CAAAAIAAJ</a>,
	    pdf:
	    <a href="http://geographiclib.sf.net/geodesic-papers/helmert80.pdf">
	      helmert80.pdf</a>.
	  <li>
	    Jacobi (1891),
	    Google id:
	    <a href="https://books.google.com/books?id=_09tAAAAMAAJ">
	      _09tAAAAMAAJ</a>,
	    pdf:
	    <a href="http://geographiclib.sf.net/geodesic-papers/jacobi-V7.pdf">
	      jacobi-V7.pdf</a>.
	</ul>
    </ol>
    Items 3&ndash;7 represent changes since the publication of the
    paper.  Changes 3&ndash;6 have been implemented in the
    <a href="index.html"> GeographicLib</a> classes
    <a href="html/classGeographicLib_1_1Geodesic.html"> Geodesic</a> and
    <a href="html/classGeographicLib_1_1GeodesicLine.html">
      GeodesicLine</a>
    in GeographicLib versions 1.27, 1.26, 1.39, and 1.25, respectively.
    Change 7 (the evaluation of the integrals in terms of elliptic
    integrals) is implemented in the <a href="index.html">
      GeographicLib</a>
    classes
    <a href="html/classGeographicLib_1_1GeodesicExact.html">
      GeodesicExact</a> and
    <a href="html/classGeographicLib_1_1GeodesicLineExact.html">
      GeodesicLineExact</a> which were added to GeographicLib 1.25.
    For geodetic applications,
    <a href="html/classGeographicLib_1_1Geodesic.html">
      Geodesic</a> and
    <a href="html/classGeographicLib_1_1GeodesicLine.html">
      GeodesicLine</a>
    are preferred, because they are about 2&ndash;3 times faster and the
    round-off errors are about 2&ndash;3 times smaller.
    <p>
      Some notes on geodesics on a <i>triaxial</i> ellipsoid are given
      in <a href="html/triaxial.html"> Geodesics on a triaxial
      ellipsoid</a>.  This examines the solution to this problem found
      by Jacobi in 1839.
    <h3>
      <a name="geodalg-errata">Errata</a> for C. F. F. Karney,
      <a href="geod.html">
	<i>Algorithms for Geodesics</i></a>,
      <a href="http://arxiv.org/abs/1109.4448v2">arXiv:1109.4448v2</a>
      (2012-03-28).
    </h3>
    <p>
      These errata apply to
      the <a href="http://arxiv.org/abs/1109.4448v2"> preprint</a>
      only:
      <ul>
	<li>
	  p. 1, col. 2, last 2 lines: replace &ldquo;present accuracy
	  and timing data are discussed&rdquo; by &ldquo;accuracy and
	  timing data are presented&rdquo;.
	<li>
	  p. 6, col. 1, last para.: replace &ldquo;Eq. (6),
	  solve&rdquo; by &ldquo;Eq. (6).  Solve&rdquo;.
	<li>
	  p. 10, col. 1, line 10: replace &ldquo;using with
	  high-precision&rdquo; by &ldquo;using high-precision&rdquo;.
	<li>
	  p. 10, col. 1, 2nd para. of Sec. 8: replace &ldquo;parallel
	  to the geodesic <i>AB</i> at <i>A</i>&rdquo; by
	  &ldquo;parallel to the geodesic <i>AB</i>
	  at <i>A</i>'&nbsp;&rdquo;.
	<li>
	  p. 10, col. 2, before Eq. (67): replace &ldquo;the curvature
	  is given by differentiating Eq. (37) with respect to &phi;
	  and dividing by&rdquo; by &ldquo;&nabla;<i>K</i> is found by
	  differentiating Eq. (37) with respect to &phi; and dividing
	  the result by&rdquo;.
      </ul>
    </p>
    <h3>
      <a name="geod-errata">Errata</a> for C. F. F. Karney,
      <a href="geod.html">
	<i>Geodesics on an ellipsoid of revolution</i></a>
      <a href="http://arxiv.org/abs/1102.1215v1">arXiv:1102.1215v1</a>
      (2011-02-07).
    </h3>
    <p>Addenda:
      <ul>
	<li>
	  Google Books does not consistently provide access to the full
	  text.  If you encounter this situation, you can download the
	  pdf files listed here:
	  <ul>
	    <li>
	      Christoffel (1910),
	      Google id:
	      <a href="https://books.google.com/books?id=9W9tAAAAMAAJ">
		9W9tAAAAMAAJ</a>,
	      pdf:
	      <a href="http://geographiclib.sf.net/geodesic-papers/christoffel-V1.pdf">
		christoffel-V1.pdf</a>.
	    <li>
	      Darboux (1894),
	      Google id:
	      <a href="https://books.google.com/books?id=hGMSAAAAIAAJ">
		hGMSAAAAIAAJ</a>,
	      pdf:
	      <a href="http://geographiclib.sf.net/geodesic-papers/darboux94.pdf">
		darboux94.pdf</a>.
	    <li>
	      Eisenhart (1909),
	      Google id:
	      <a href="https://books.google.com/books?id=hkENAAAAYAAJ">
		hkENAAAAYAAJ</a>,
	      pdf:
	      <a href="http://geographiclib.sf.net/geodesic-papers/eisenhart09.pdf">
		eisenhart09.pdf</a>.
	    <li>
	      Forsyth (1896),
	      Google id:
	      <a href="https://books.google.com/books?id=YsAKAAAAIAAJ">
		YsAKAAAAIAAJ</a>,
	      pdf:
	      <a href="http://geographiclib.sf.net/geodesic-papers/messmath-V25.pdf">
		messmath-V25.pdf</a>.
	    <li>
	      Gauss (1902),
	      Google id:
	      <a href="https://books.google.com/books?id=a1wTJR3kHwUC">
		a1wTJR3kHwUC</a>,
	      pdf:
	      <a href="http://geographiclib.sf.net/geodesic-papers/gauss28-en.pdf">
		gauss28-en.pdf</a>.
	    <li>
	      Gauss (1903),
	      Google id:
	      <a href="https://books.google.com/books?id=ICwPAAAAIAAJ">
		ICwPAAAAIAAJ</a>,
	      pdf:
	      <a href="http://geographiclib.sf.net/geodesic-papers/gauss-V9.pdf">
		gauss-V9.pdf</a>.
	    <li>
	      Helmert (1880),
	      Google id:
	      <a href="https://books.google.com/books?id=qt2CAAAAIAAJ">
		qt2CAAAAIAAJ</a>,
	      pdf:
	      <a href="http://geographiclib.sf.net/geodesic-papers/helmert80.pdf">
		helmert80.pdf</a>.
	    <li>
	      Jacobi (1891),
	      Google id:
	      <a href="https://books.google.com/books?id=_09tAAAAMAAJ">
		_09tAAAAMAAJ</a>,
	      pdf:
	      <a href="http://geographiclib.sf.net/geodesic-papers/jacobi-V7.pdf">
		jacobi-V7.pdf</a>.
	  </ul>
      </ul>
    <p>Errata:
      <ul>
	<li>
	  Sec. 1, para. 1, line 7: replace &ldquo;taking&rdquo; by
	  &ldquo;taken&rdquo;.
	<li>
	  Sec. 1, last para., line 4: replace &ldquo;concerned
	  the&rdquo; by &ldquo;concerned with the&rdquo;.
	<li>
	  p. 5, Eq. (44): replace &ldquo;2<i>l</i>,&sigma;&rdquo; by
	  &ldquo;2<i>l</i>&sigma;,&rdquo;.
	<li>
	  Sec. 5, para. 1, line 1: replace &ldquo;simply matter&rdquo;
	  by &ldquo;simply a matter&rdquo;.
	<li>
	  p. 5, col. 2, para. 1, 2nd last line: replace &ldquo;gave him
	  with a&rdquo; by &ldquo;gave him a&rdquo;.
	<li>
	  p. 6, col. 2, following Eq. (53): replace &ldquo;as noted as
	  the end&rdquo; by &ldquo;as noted at the end&rdquo;.
	<li>
	  p. 8, col. 1, para. 2, line 3: replace &ldquo;the pair the
	  pair&rdquo; by &ldquo;the pair&rdquo;.
	<li>
	  Fig. 4, 2nd line of caption: delete &ldquo;close&rdquo;.
	<li>
	  p. 9, col. 1, 2nd last line: replace &ldquo;loose&rdquo; by
	  &ldquo;lose&rdquo;.
	<li>
	  p. 9, col. 2, Eq. (64): Helmert (1880), Eq. (7.3.7), also
	  suggested using this equation to solve for
	  &alpha;<sub>1</sub>.
	<li>
	  Sec. 7, 2nd last para., last line: replace &ldquo;Helmert
	  (1880, &sect;9.2)&rdquo; by &ldquo;Helmert (1880,
	  &sect;7.2)&rdquo;.
	<li>
	  Sec. 7: remove the last sentence; there's an updated version
	  of Rapp (1993) available
	  at <a href="http://hdl.handle.net/1811/24409">
	  http://hdl.handle.net/1811/24409</a>.
	<li>
	  p. 12, col. 1, Eq. (73): this equation may be obtained from
	  Helmert (1880), Eq. (6.9.8b).
	<li>
	  p. 13, col. 1, para. 2, line 3: replace
	  &ldquo;&phi;<sub>1</sub> + &phi;<sub>1</sub> = 0&rdquo; by
	  &ldquo;&phi;<sub>1</sub> + &phi;<sub>2</sub> = 0&rdquo;.
	<li>
	  Sec. 11, last para., line 1: replace &ldquo;solution&rdquo; by
	  &ldquo;solutions&rdquo;.
	<li>
	  Sec. 11, last line: replace &ldquo;slowly that&rdquo; by
	  &ldquo;slowly than&rdquo;.
	<li>
	  p. 19, col. 2, line 6: replace &ldquo;Thus leads to&rdquo; by
	  &ldquo;This leads to&rdquo;.
	<li>
	  p. 20, col. 2, 4 lines before 2nd eq.: replace
	  &ldquo;solving&rdquo; by &ldquo;solved&rdquo;.
	<li>
	  Sec. 14, line 4: replace &ldquo;these rule&rdquo; by
	  &ldquo;these rules&rdquo;.
	<li>
	  p. 20, col. 2, line 5: replace &ldquo;of of&rdquo; by
	  &ldquo;of&rdquo;.
	<li>
	  p. 21, col. 2, 9th last line: replace &ldquo;the
	  point with is a distance&rdquo; by &ldquo;the point which is a
	  distance&rdquo;.
	<li>
	  p. 21, col. 2, 3rd last line: replace &ldquo;it can also be
	  solving&rdquo; by &ldquo;it can also be solved&rdquo;.
	<li>
	  p. 23, col. 1, para. 2, line 3: replace
	  &ldquo;2&pi;<i>c</i><sup>2</sup>&rdquo; by
	  &ldquo;2&pi;<i>R<sub>q</sub></i><sup>2</sup>&rdquo;.
	<li>
	  p. 23, col. 1, para. 2, 4th last line: replace
	  &ldquo;the Japan&rdquo; by &ldquo;Japan&rdquo;.
	<li>
	  p. 23, col. 2, last line: replace &ldquo;has has&rdquo; by
	  &ldquo;has&rdquo;.
	<li>
	  App. B, line 7: replace &ldquo;can therefore by used&rdquo; by
	  &ldquo;can therefore be used&rdquo;.
	<li>
	  p. 25, col. 1, 2nd last line: the link to Olver et al.,
	  Sec. 1.11(iii), is incorrect; it should be
	  <a href="http://dlmf.nist.gov/1.11.iii">
	  http://dlmf.nist.gov/1.11.iii</a>.
	<li>
	  p. 25, col. 2, last line: replace &ldquo;<i>x</i>
	  &le; <i>e</i><sup>2</sup>&rdquo; by &ldquo;|<i>x</i>|
	  &le; <i>e</i><sup>2</sup>&rdquo;.
	<li>
	  p. 26, col. 1, line 2 following Eq. (C1): replace
	  &ldquo;<i>AFGB</i>&rdquo; by &ldquo;<i>AFHB</i>&rdquo;.
	<li>
	  p. 26, col. 2, following 3rd eq.: insert comma between the
	  inline equations for <i>b</i> and &gamma;.
	<li>
	  p. 27, col. 2, Eq. (D4): replace
	  &ldquo;<i>c</i><sup>2</sup>&rdquo; by
	  &ldquo;<i>R<sub>q</sub></i><sup>2</sup>&rdquo;.
	<li>
	  p. 29, col. 1, refs. Oriani (1806, 1808, 1810): replace
	  &ldquo;trigonemetria&rdquo; by &ldquo;trigonometria&rdquo;.
      </ul>
    </p>
    <a href="geod.html">Back to resource page for geodesics.</a>
    <hr>
    <address>Charles Karney
      <a href="mailto:charles@karney.com">&lt;charles@karney.com&gt;</a>
      (2015-05-09)</address>
    <br>
    <a href="http://geographiclib.sourceforge.net">
      GeographicLib home
    </a>
  </body>
</html>
